We study the integrability properties of the gain part of the Boltzmann collision operator using radial symmetrization techniques from harmonic analysis to show Young's inequality in the case of hard potentials and Hardy-Littlewood-Sobolev inequality for soft potentials. The contacts are given by exact formulas depending on the angular cross section. By applying these estimates we can revisit and obtain new results for existence and uniqueness to the corresponding space inhomogeneous equations with special initial data. This is work partly in collaboration with E. Carneiro and R. Alonso